MAYBE * Step 1: DependencyPairs MAYBE + Considered Problem: - Strict TRS: a__2nd(X) -> 2nd(X) a__2nd(cons(X,XS)) -> a__head(mark(XS)) a__from(X) -> cons(mark(X),from(s(X))) a__from(X) -> from(X) a__head(X) -> head(X) a__head(cons(X,XS)) -> mark(X) a__sel(X1,X2) -> sel(X1,X2) a__sel(0(),cons(X,XS)) -> mark(X) a__sel(s(N),cons(X,XS)) -> a__sel(mark(N),mark(XS)) a__take(X1,X2) -> take(X1,X2) a__take(0(),XS) -> nil() a__take(s(N),cons(X,XS)) -> cons(mark(X),take(N,XS)) mark(0()) -> 0() mark(2nd(X)) -> a__2nd(mark(X)) mark(cons(X1,X2)) -> cons(mark(X1),X2) mark(from(X)) -> a__from(mark(X)) mark(head(X)) -> a__head(mark(X)) mark(nil()) -> nil() mark(s(X)) -> s(mark(X)) mark(sel(X1,X2)) -> a__sel(mark(X1),mark(X2)) mark(take(X1,X2)) -> a__take(mark(X1),mark(X2)) - Signature: {a__2nd/1,a__from/1,a__head/1,a__sel/2,a__take/2,mark/1} / {0/0,2nd/1,cons/2,from/1,head/1,nil/0,s/1,sel/2 ,take/2} - Obligation: innermost runtime complexity wrt. defined symbols {a__2nd,a__from,a__head,a__sel,a__take ,mark} and constructors {0,2nd,cons,from,head,nil,s,sel,take} + Applied Processor: DependencyPairs {dpKind_ = DT} + Details: We add the following dependency tuples: Strict DPs a__2nd#(X) -> c_1() a__2nd#(cons(X,XS)) -> c_2(a__head#(mark(XS)),mark#(XS)) a__from#(X) -> c_3(mark#(X)) a__from#(X) -> c_4() a__head#(X) -> c_5() a__head#(cons(X,XS)) -> c_6(mark#(X)) a__sel#(X1,X2) -> c_7() a__sel#(0(),cons(X,XS)) -> c_8(mark#(X)) a__sel#(s(N),cons(X,XS)) -> c_9(a__sel#(mark(N),mark(XS)),mark#(N),mark#(XS)) a__take#(X1,X2) -> c_10() a__take#(0(),XS) -> c_11() a__take#(s(N),cons(X,XS)) -> c_12(mark#(X)) mark#(0()) -> c_13() mark#(2nd(X)) -> c_14(a__2nd#(mark(X)),mark#(X)) mark#(cons(X1,X2)) -> c_15(mark#(X1)) mark#(from(X)) -> c_16(a__from#(mark(X)),mark#(X)) mark#(head(X)) -> c_17(a__head#(mark(X)),mark#(X)) mark#(nil()) -> c_18() mark#(s(X)) -> c_19(mark#(X)) mark#(sel(X1,X2)) -> c_20(a__sel#(mark(X1),mark(X2)),mark#(X1),mark#(X2)) mark#(take(X1,X2)) -> c_21(a__take#(mark(X1),mark(X2)),mark#(X1),mark#(X2)) Weak DPs and mark the set of starting terms. * Step 2: PredecessorEstimation MAYBE + Considered Problem: - Strict DPs: a__2nd#(X) -> c_1() a__2nd#(cons(X,XS)) -> c_2(a__head#(mark(XS)),mark#(XS)) a__from#(X) -> c_3(mark#(X)) a__from#(X) -> c_4() a__head#(X) -> c_5() a__head#(cons(X,XS)) -> c_6(mark#(X)) a__sel#(X1,X2) -> c_7() a__sel#(0(),cons(X,XS)) -> c_8(mark#(X)) a__sel#(s(N),cons(X,XS)) -> c_9(a__sel#(mark(N),mark(XS)),mark#(N),mark#(XS)) a__take#(X1,X2) -> c_10() a__take#(0(),XS) -> c_11() a__take#(s(N),cons(X,XS)) -> c_12(mark#(X)) mark#(0()) -> c_13() mark#(2nd(X)) -> c_14(a__2nd#(mark(X)),mark#(X)) mark#(cons(X1,X2)) -> c_15(mark#(X1)) mark#(from(X)) -> c_16(a__from#(mark(X)),mark#(X)) mark#(head(X)) -> c_17(a__head#(mark(X)),mark#(X)) mark#(nil()) -> c_18() mark#(s(X)) -> c_19(mark#(X)) mark#(sel(X1,X2)) -> c_20(a__sel#(mark(X1),mark(X2)),mark#(X1),mark#(X2)) mark#(take(X1,X2)) -> c_21(a__take#(mark(X1),mark(X2)),mark#(X1),mark#(X2)) - Weak TRS: a__2nd(X) -> 2nd(X) a__2nd(cons(X,XS)) -> a__head(mark(XS)) a__from(X) -> cons(mark(X),from(s(X))) a__from(X) -> from(X) a__head(X) -> head(X) a__head(cons(X,XS)) -> mark(X) a__sel(X1,X2) -> sel(X1,X2) a__sel(0(),cons(X,XS)) -> mark(X) a__sel(s(N),cons(X,XS)) -> a__sel(mark(N),mark(XS)) a__take(X1,X2) -> take(X1,X2) a__take(0(),XS) -> nil() a__take(s(N),cons(X,XS)) -> cons(mark(X),take(N,XS)) mark(0()) -> 0() mark(2nd(X)) -> a__2nd(mark(X)) mark(cons(X1,X2)) -> cons(mark(X1),X2) mark(from(X)) -> a__from(mark(X)) mark(head(X)) -> a__head(mark(X)) mark(nil()) -> nil() mark(s(X)) -> s(mark(X)) mark(sel(X1,X2)) -> a__sel(mark(X1),mark(X2)) mark(take(X1,X2)) -> a__take(mark(X1),mark(X2)) - Signature: {a__2nd/1,a__from/1,a__head/1,a__sel/2,a__take/2,mark/1,a__2nd#/1,a__from#/1,a__head#/1,a__sel#/2,a__take#/2 ,mark#/1} / {0/0,2nd/1,cons/2,from/1,head/1,nil/0,s/1,sel/2,take/2,c_1/0,c_2/2,c_3/1,c_4/0,c_5/0,c_6/1,c_7/0 ,c_8/1,c_9/3,c_10/0,c_11/0,c_12/1,c_13/0,c_14/2,c_15/1,c_16/2,c_17/2,c_18/0,c_19/1,c_20/3,c_21/3} - Obligation: innermost runtime complexity wrt. defined symbols {a__2nd#,a__from#,a__head#,a__sel#,a__take# ,mark#} and constructors {0,2nd,cons,from,head,nil,s,sel,take} + Applied Processor: PredecessorEstimation {onSelection = all simple predecessor estimation selector} + Details: We estimate the number of application of {1,4,5,7,10,11,13,18} by application of Pre({1,4,5,7,10,11,13,18}) = {2,3,6,8,9,12,14,15,16,17,19,20,21}. Here rules are labelled as follows: 1: a__2nd#(X) -> c_1() 2: a__2nd#(cons(X,XS)) -> c_2(a__head#(mark(XS)),mark#(XS)) 3: a__from#(X) -> c_3(mark#(X)) 4: a__from#(X) -> c_4() 5: a__head#(X) -> c_5() 6: a__head#(cons(X,XS)) -> c_6(mark#(X)) 7: a__sel#(X1,X2) -> c_7() 8: a__sel#(0(),cons(X,XS)) -> c_8(mark#(X)) 9: a__sel#(s(N),cons(X,XS)) -> c_9(a__sel#(mark(N),mark(XS)),mark#(N),mark#(XS)) 10: a__take#(X1,X2) -> c_10() 11: a__take#(0(),XS) -> c_11() 12: a__take#(s(N),cons(X,XS)) -> c_12(mark#(X)) 13: mark#(0()) -> c_13() 14: mark#(2nd(X)) -> c_14(a__2nd#(mark(X)),mark#(X)) 15: mark#(cons(X1,X2)) -> c_15(mark#(X1)) 16: mark#(from(X)) -> c_16(a__from#(mark(X)),mark#(X)) 17: mark#(head(X)) -> c_17(a__head#(mark(X)),mark#(X)) 18: mark#(nil()) -> c_18() 19: mark#(s(X)) -> c_19(mark#(X)) 20: mark#(sel(X1,X2)) -> c_20(a__sel#(mark(X1),mark(X2)),mark#(X1),mark#(X2)) 21: mark#(take(X1,X2)) -> c_21(a__take#(mark(X1),mark(X2)),mark#(X1),mark#(X2)) * Step 3: RemoveWeakSuffixes MAYBE + Considered Problem: - Strict DPs: a__2nd#(cons(X,XS)) -> c_2(a__head#(mark(XS)),mark#(XS)) a__from#(X) -> c_3(mark#(X)) a__head#(cons(X,XS)) -> c_6(mark#(X)) a__sel#(0(),cons(X,XS)) -> c_8(mark#(X)) a__sel#(s(N),cons(X,XS)) -> c_9(a__sel#(mark(N),mark(XS)),mark#(N),mark#(XS)) a__take#(s(N),cons(X,XS)) -> c_12(mark#(X)) mark#(2nd(X)) -> c_14(a__2nd#(mark(X)),mark#(X)) mark#(cons(X1,X2)) -> c_15(mark#(X1)) mark#(from(X)) -> c_16(a__from#(mark(X)),mark#(X)) mark#(head(X)) -> c_17(a__head#(mark(X)),mark#(X)) mark#(s(X)) -> c_19(mark#(X)) mark#(sel(X1,X2)) -> c_20(a__sel#(mark(X1),mark(X2)),mark#(X1),mark#(X2)) mark#(take(X1,X2)) -> c_21(a__take#(mark(X1),mark(X2)),mark#(X1),mark#(X2)) - Weak DPs: a__2nd#(X) -> c_1() a__from#(X) -> c_4() a__head#(X) -> c_5() a__sel#(X1,X2) -> c_7() a__take#(X1,X2) -> c_10() a__take#(0(),XS) -> c_11() mark#(0()) -> c_13() mark#(nil()) -> c_18() - Weak TRS: a__2nd(X) -> 2nd(X) a__2nd(cons(X,XS)) -> a__head(mark(XS)) a__from(X) -> cons(mark(X),from(s(X))) a__from(X) -> from(X) a__head(X) -> head(X) a__head(cons(X,XS)) -> mark(X) a__sel(X1,X2) -> sel(X1,X2) a__sel(0(),cons(X,XS)) -> mark(X) a__sel(s(N),cons(X,XS)) -> a__sel(mark(N),mark(XS)) a__take(X1,X2) -> take(X1,X2) a__take(0(),XS) -> nil() a__take(s(N),cons(X,XS)) -> cons(mark(X),take(N,XS)) mark(0()) -> 0() mark(2nd(X)) -> a__2nd(mark(X)) mark(cons(X1,X2)) -> cons(mark(X1),X2) mark(from(X)) -> a__from(mark(X)) mark(head(X)) -> a__head(mark(X)) mark(nil()) -> nil() mark(s(X)) -> s(mark(X)) mark(sel(X1,X2)) -> a__sel(mark(X1),mark(X2)) mark(take(X1,X2)) -> a__take(mark(X1),mark(X2)) - Signature: {a__2nd/1,a__from/1,a__head/1,a__sel/2,a__take/2,mark/1,a__2nd#/1,a__from#/1,a__head#/1,a__sel#/2,a__take#/2 ,mark#/1} / {0/0,2nd/1,cons/2,from/1,head/1,nil/0,s/1,sel/2,take/2,c_1/0,c_2/2,c_3/1,c_4/0,c_5/0,c_6/1,c_7/0 ,c_8/1,c_9/3,c_10/0,c_11/0,c_12/1,c_13/0,c_14/2,c_15/1,c_16/2,c_17/2,c_18/0,c_19/1,c_20/3,c_21/3} - Obligation: innermost runtime complexity wrt. defined symbols {a__2nd#,a__from#,a__head#,a__sel#,a__take# ,mark#} and constructors {0,2nd,cons,from,head,nil,s,sel,take} + Applied Processor: RemoveWeakSuffixes + Details: Consider the dependency graph 1:S:a__2nd#(cons(X,XS)) -> c_2(a__head#(mark(XS)),mark#(XS)) -->_2 mark#(take(X1,X2)) -> c_21(a__take#(mark(X1),mark(X2)),mark#(X1),mark#(X2)):13 -->_2 mark#(sel(X1,X2)) -> c_20(a__sel#(mark(X1),mark(X2)),mark#(X1),mark#(X2)):12 -->_2 mark#(s(X)) -> c_19(mark#(X)):11 -->_2 mark#(head(X)) -> c_17(a__head#(mark(X)),mark#(X)):10 -->_2 mark#(from(X)) -> c_16(a__from#(mark(X)),mark#(X)):9 -->_2 mark#(cons(X1,X2)) -> c_15(mark#(X1)):8 -->_2 mark#(2nd(X)) -> c_14(a__2nd#(mark(X)),mark#(X)):7 -->_1 a__head#(cons(X,XS)) -> c_6(mark#(X)):3 -->_2 mark#(nil()) -> c_18():21 -->_2 mark#(0()) -> c_13():20 -->_1 a__head#(X) -> c_5():16 2:S:a__from#(X) -> c_3(mark#(X)) -->_1 mark#(take(X1,X2)) -> c_21(a__take#(mark(X1),mark(X2)),mark#(X1),mark#(X2)):13 -->_1 mark#(sel(X1,X2)) -> c_20(a__sel#(mark(X1),mark(X2)),mark#(X1),mark#(X2)):12 -->_1 mark#(s(X)) -> c_19(mark#(X)):11 -->_1 mark#(head(X)) -> c_17(a__head#(mark(X)),mark#(X)):10 -->_1 mark#(from(X)) -> c_16(a__from#(mark(X)),mark#(X)):9 -->_1 mark#(cons(X1,X2)) -> c_15(mark#(X1)):8 -->_1 mark#(2nd(X)) -> c_14(a__2nd#(mark(X)),mark#(X)):7 -->_1 mark#(nil()) -> c_18():21 -->_1 mark#(0()) -> c_13():20 3:S:a__head#(cons(X,XS)) -> c_6(mark#(X)) -->_1 mark#(take(X1,X2)) -> c_21(a__take#(mark(X1),mark(X2)),mark#(X1),mark#(X2)):13 -->_1 mark#(sel(X1,X2)) -> c_20(a__sel#(mark(X1),mark(X2)),mark#(X1),mark#(X2)):12 -->_1 mark#(s(X)) -> c_19(mark#(X)):11 -->_1 mark#(head(X)) -> c_17(a__head#(mark(X)),mark#(X)):10 -->_1 mark#(from(X)) -> c_16(a__from#(mark(X)),mark#(X)):9 -->_1 mark#(cons(X1,X2)) -> c_15(mark#(X1)):8 -->_1 mark#(2nd(X)) -> c_14(a__2nd#(mark(X)),mark#(X)):7 -->_1 mark#(nil()) -> c_18():21 -->_1 mark#(0()) -> c_13():20 4:S:a__sel#(0(),cons(X,XS)) -> c_8(mark#(X)) -->_1 mark#(take(X1,X2)) -> c_21(a__take#(mark(X1),mark(X2)),mark#(X1),mark#(X2)):13 -->_1 mark#(sel(X1,X2)) -> c_20(a__sel#(mark(X1),mark(X2)),mark#(X1),mark#(X2)):12 -->_1 mark#(s(X)) -> c_19(mark#(X)):11 -->_1 mark#(head(X)) -> c_17(a__head#(mark(X)),mark#(X)):10 -->_1 mark#(from(X)) -> c_16(a__from#(mark(X)),mark#(X)):9 -->_1 mark#(cons(X1,X2)) -> c_15(mark#(X1)):8 -->_1 mark#(2nd(X)) -> c_14(a__2nd#(mark(X)),mark#(X)):7 -->_1 mark#(nil()) -> c_18():21 -->_1 mark#(0()) -> c_13():20 5:S:a__sel#(s(N),cons(X,XS)) -> c_9(a__sel#(mark(N),mark(XS)),mark#(N),mark#(XS)) -->_3 mark#(take(X1,X2)) -> c_21(a__take#(mark(X1),mark(X2)),mark#(X1),mark#(X2)):13 -->_2 mark#(take(X1,X2)) -> c_21(a__take#(mark(X1),mark(X2)),mark#(X1),mark#(X2)):13 -->_3 mark#(sel(X1,X2)) -> c_20(a__sel#(mark(X1),mark(X2)),mark#(X1),mark#(X2)):12 -->_2 mark#(sel(X1,X2)) -> c_20(a__sel#(mark(X1),mark(X2)),mark#(X1),mark#(X2)):12 -->_3 mark#(s(X)) -> c_19(mark#(X)):11 -->_2 mark#(s(X)) -> c_19(mark#(X)):11 -->_3 mark#(head(X)) -> c_17(a__head#(mark(X)),mark#(X)):10 -->_2 mark#(head(X)) -> c_17(a__head#(mark(X)),mark#(X)):10 -->_3 mark#(from(X)) -> c_16(a__from#(mark(X)),mark#(X)):9 -->_2 mark#(from(X)) -> c_16(a__from#(mark(X)),mark#(X)):9 -->_3 mark#(cons(X1,X2)) -> c_15(mark#(X1)):8 -->_2 mark#(cons(X1,X2)) -> c_15(mark#(X1)):8 -->_3 mark#(2nd(X)) -> c_14(a__2nd#(mark(X)),mark#(X)):7 -->_2 mark#(2nd(X)) -> c_14(a__2nd#(mark(X)),mark#(X)):7 -->_3 mark#(nil()) -> c_18():21 -->_2 mark#(nil()) -> c_18():21 -->_3 mark#(0()) -> c_13():20 -->_2 mark#(0()) -> c_13():20 -->_1 a__sel#(X1,X2) -> c_7():17 -->_1 a__sel#(s(N),cons(X,XS)) -> c_9(a__sel#(mark(N),mark(XS)),mark#(N),mark#(XS)):5 -->_1 a__sel#(0(),cons(X,XS)) -> c_8(mark#(X)):4 6:S:a__take#(s(N),cons(X,XS)) -> c_12(mark#(X)) -->_1 mark#(take(X1,X2)) -> c_21(a__take#(mark(X1),mark(X2)),mark#(X1),mark#(X2)):13 -->_1 mark#(sel(X1,X2)) -> c_20(a__sel#(mark(X1),mark(X2)),mark#(X1),mark#(X2)):12 -->_1 mark#(s(X)) -> c_19(mark#(X)):11 -->_1 mark#(head(X)) -> c_17(a__head#(mark(X)),mark#(X)):10 -->_1 mark#(from(X)) -> c_16(a__from#(mark(X)),mark#(X)):9 -->_1 mark#(cons(X1,X2)) -> c_15(mark#(X1)):8 -->_1 mark#(2nd(X)) -> c_14(a__2nd#(mark(X)),mark#(X)):7 -->_1 mark#(nil()) -> c_18():21 -->_1 mark#(0()) -> c_13():20 7:S:mark#(2nd(X)) -> c_14(a__2nd#(mark(X)),mark#(X)) -->_2 mark#(take(X1,X2)) -> c_21(a__take#(mark(X1),mark(X2)),mark#(X1),mark#(X2)):13 -->_2 mark#(sel(X1,X2)) -> c_20(a__sel#(mark(X1),mark(X2)),mark#(X1),mark#(X2)):12 -->_2 mark#(s(X)) -> c_19(mark#(X)):11 -->_2 mark#(head(X)) -> c_17(a__head#(mark(X)),mark#(X)):10 -->_2 mark#(from(X)) -> c_16(a__from#(mark(X)),mark#(X)):9 -->_2 mark#(cons(X1,X2)) -> c_15(mark#(X1)):8 -->_2 mark#(nil()) -> c_18():21 -->_2 mark#(0()) -> c_13():20 -->_1 a__2nd#(X) -> c_1():14 -->_2 mark#(2nd(X)) -> c_14(a__2nd#(mark(X)),mark#(X)):7 -->_1 a__2nd#(cons(X,XS)) -> c_2(a__head#(mark(XS)),mark#(XS)):1 8:S:mark#(cons(X1,X2)) -> c_15(mark#(X1)) -->_1 mark#(take(X1,X2)) -> c_21(a__take#(mark(X1),mark(X2)),mark#(X1),mark#(X2)):13 -->_1 mark#(sel(X1,X2)) -> c_20(a__sel#(mark(X1),mark(X2)),mark#(X1),mark#(X2)):12 -->_1 mark#(s(X)) -> c_19(mark#(X)):11 -->_1 mark#(head(X)) -> c_17(a__head#(mark(X)),mark#(X)):10 -->_1 mark#(from(X)) -> c_16(a__from#(mark(X)),mark#(X)):9 -->_1 mark#(nil()) -> c_18():21 -->_1 mark#(0()) -> c_13():20 -->_1 mark#(cons(X1,X2)) -> c_15(mark#(X1)):8 -->_1 mark#(2nd(X)) -> c_14(a__2nd#(mark(X)),mark#(X)):7 9:S:mark#(from(X)) -> c_16(a__from#(mark(X)),mark#(X)) -->_2 mark#(take(X1,X2)) -> c_21(a__take#(mark(X1),mark(X2)),mark#(X1),mark#(X2)):13 -->_2 mark#(sel(X1,X2)) -> c_20(a__sel#(mark(X1),mark(X2)),mark#(X1),mark#(X2)):12 -->_2 mark#(s(X)) -> c_19(mark#(X)):11 -->_2 mark#(head(X)) -> c_17(a__head#(mark(X)),mark#(X)):10 -->_2 mark#(nil()) -> c_18():21 -->_2 mark#(0()) -> c_13():20 -->_1 a__from#(X) -> c_4():15 -->_2 mark#(from(X)) -> c_16(a__from#(mark(X)),mark#(X)):9 -->_2 mark#(cons(X1,X2)) -> c_15(mark#(X1)):8 -->_2 mark#(2nd(X)) -> c_14(a__2nd#(mark(X)),mark#(X)):7 -->_1 a__from#(X) -> c_3(mark#(X)):2 10:S:mark#(head(X)) -> c_17(a__head#(mark(X)),mark#(X)) -->_2 mark#(take(X1,X2)) -> c_21(a__take#(mark(X1),mark(X2)),mark#(X1),mark#(X2)):13 -->_2 mark#(sel(X1,X2)) -> c_20(a__sel#(mark(X1),mark(X2)),mark#(X1),mark#(X2)):12 -->_2 mark#(s(X)) -> c_19(mark#(X)):11 -->_2 mark#(nil()) -> c_18():21 -->_2 mark#(0()) -> c_13():20 -->_1 a__head#(X) -> c_5():16 -->_2 mark#(head(X)) -> c_17(a__head#(mark(X)),mark#(X)):10 -->_2 mark#(from(X)) -> c_16(a__from#(mark(X)),mark#(X)):9 -->_2 mark#(cons(X1,X2)) -> c_15(mark#(X1)):8 -->_2 mark#(2nd(X)) -> c_14(a__2nd#(mark(X)),mark#(X)):7 -->_1 a__head#(cons(X,XS)) -> c_6(mark#(X)):3 11:S:mark#(s(X)) -> c_19(mark#(X)) -->_1 mark#(take(X1,X2)) -> c_21(a__take#(mark(X1),mark(X2)),mark#(X1),mark#(X2)):13 -->_1 mark#(sel(X1,X2)) -> c_20(a__sel#(mark(X1),mark(X2)),mark#(X1),mark#(X2)):12 -->_1 mark#(nil()) -> c_18():21 -->_1 mark#(0()) -> c_13():20 -->_1 mark#(s(X)) -> c_19(mark#(X)):11 -->_1 mark#(head(X)) -> c_17(a__head#(mark(X)),mark#(X)):10 -->_1 mark#(from(X)) -> c_16(a__from#(mark(X)),mark#(X)):9 -->_1 mark#(cons(X1,X2)) -> c_15(mark#(X1)):8 -->_1 mark#(2nd(X)) -> c_14(a__2nd#(mark(X)),mark#(X)):7 12:S:mark#(sel(X1,X2)) -> c_20(a__sel#(mark(X1),mark(X2)),mark#(X1),mark#(X2)) -->_3 mark#(take(X1,X2)) -> c_21(a__take#(mark(X1),mark(X2)),mark#(X1),mark#(X2)):13 -->_2 mark#(take(X1,X2)) -> c_21(a__take#(mark(X1),mark(X2)),mark#(X1),mark#(X2)):13 -->_3 mark#(nil()) -> c_18():21 -->_2 mark#(nil()) -> c_18():21 -->_3 mark#(0()) -> c_13():20 -->_2 mark#(0()) -> c_13():20 -->_1 a__sel#(X1,X2) -> c_7():17 -->_3 mark#(sel(X1,X2)) -> c_20(a__sel#(mark(X1),mark(X2)),mark#(X1),mark#(X2)):12 -->_2 mark#(sel(X1,X2)) -> c_20(a__sel#(mark(X1),mark(X2)),mark#(X1),mark#(X2)):12 -->_3 mark#(s(X)) -> c_19(mark#(X)):11 -->_2 mark#(s(X)) -> c_19(mark#(X)):11 -->_3 mark#(head(X)) -> c_17(a__head#(mark(X)),mark#(X)):10 -->_2 mark#(head(X)) -> c_17(a__head#(mark(X)),mark#(X)):10 -->_3 mark#(from(X)) -> c_16(a__from#(mark(X)),mark#(X)):9 -->_2 mark#(from(X)) -> c_16(a__from#(mark(X)),mark#(X)):9 -->_3 mark#(cons(X1,X2)) -> c_15(mark#(X1)):8 -->_2 mark#(cons(X1,X2)) -> c_15(mark#(X1)):8 -->_3 mark#(2nd(X)) -> c_14(a__2nd#(mark(X)),mark#(X)):7 -->_2 mark#(2nd(X)) -> c_14(a__2nd#(mark(X)),mark#(X)):7 -->_1 a__sel#(s(N),cons(X,XS)) -> c_9(a__sel#(mark(N),mark(XS)),mark#(N),mark#(XS)):5 -->_1 a__sel#(0(),cons(X,XS)) -> c_8(mark#(X)):4 13:S:mark#(take(X1,X2)) -> c_21(a__take#(mark(X1),mark(X2)),mark#(X1),mark#(X2)) -->_3 mark#(nil()) -> c_18():21 -->_2 mark#(nil()) -> c_18():21 -->_3 mark#(0()) -> c_13():20 -->_2 mark#(0()) -> c_13():20 -->_1 a__take#(0(),XS) -> c_11():19 -->_1 a__take#(X1,X2) -> c_10():18 -->_3 mark#(take(X1,X2)) -> c_21(a__take#(mark(X1),mark(X2)),mark#(X1),mark#(X2)):13 -->_2 mark#(take(X1,X2)) -> c_21(a__take#(mark(X1),mark(X2)),mark#(X1),mark#(X2)):13 -->_3 mark#(sel(X1,X2)) -> c_20(a__sel#(mark(X1),mark(X2)),mark#(X1),mark#(X2)):12 -->_2 mark#(sel(X1,X2)) -> c_20(a__sel#(mark(X1),mark(X2)),mark#(X1),mark#(X2)):12 -->_3 mark#(s(X)) -> c_19(mark#(X)):11 -->_2 mark#(s(X)) -> c_19(mark#(X)):11 -->_3 mark#(head(X)) -> c_17(a__head#(mark(X)),mark#(X)):10 -->_2 mark#(head(X)) -> c_17(a__head#(mark(X)),mark#(X)):10 -->_3 mark#(from(X)) -> c_16(a__from#(mark(X)),mark#(X)):9 -->_2 mark#(from(X)) -> c_16(a__from#(mark(X)),mark#(X)):9 -->_3 mark#(cons(X1,X2)) -> c_15(mark#(X1)):8 -->_2 mark#(cons(X1,X2)) -> c_15(mark#(X1)):8 -->_3 mark#(2nd(X)) -> c_14(a__2nd#(mark(X)),mark#(X)):7 -->_2 mark#(2nd(X)) -> c_14(a__2nd#(mark(X)),mark#(X)):7 -->_1 a__take#(s(N),cons(X,XS)) -> c_12(mark#(X)):6 14:W:a__2nd#(X) -> c_1() 15:W:a__from#(X) -> c_4() 16:W:a__head#(X) -> c_5() 17:W:a__sel#(X1,X2) -> c_7() 18:W:a__take#(X1,X2) -> c_10() 19:W:a__take#(0(),XS) -> c_11() 20:W:mark#(0()) -> c_13() 21:W:mark#(nil()) -> c_18() The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed. 14: a__2nd#(X) -> c_1() 15: a__from#(X) -> c_4() 16: a__head#(X) -> c_5() 17: a__sel#(X1,X2) -> c_7() 18: a__take#(X1,X2) -> c_10() 19: a__take#(0(),XS) -> c_11() 20: mark#(0()) -> c_13() 21: mark#(nil()) -> c_18() * Step 4: Failure MAYBE + Considered Problem: - Strict DPs: a__2nd#(cons(X,XS)) -> c_2(a__head#(mark(XS)),mark#(XS)) a__from#(X) -> c_3(mark#(X)) a__head#(cons(X,XS)) -> c_6(mark#(X)) a__sel#(0(),cons(X,XS)) -> c_8(mark#(X)) a__sel#(s(N),cons(X,XS)) -> c_9(a__sel#(mark(N),mark(XS)),mark#(N),mark#(XS)) a__take#(s(N),cons(X,XS)) -> c_12(mark#(X)) mark#(2nd(X)) -> c_14(a__2nd#(mark(X)),mark#(X)) mark#(cons(X1,X2)) -> c_15(mark#(X1)) mark#(from(X)) -> c_16(a__from#(mark(X)),mark#(X)) mark#(head(X)) -> c_17(a__head#(mark(X)),mark#(X)) mark#(s(X)) -> c_19(mark#(X)) mark#(sel(X1,X2)) -> c_20(a__sel#(mark(X1),mark(X2)),mark#(X1),mark#(X2)) mark#(take(X1,X2)) -> c_21(a__take#(mark(X1),mark(X2)),mark#(X1),mark#(X2)) - Weak TRS: a__2nd(X) -> 2nd(X) a__2nd(cons(X,XS)) -> a__head(mark(XS)) a__from(X) -> cons(mark(X),from(s(X))) a__from(X) -> from(X) a__head(X) -> head(X) a__head(cons(X,XS)) -> mark(X) a__sel(X1,X2) -> sel(X1,X2) a__sel(0(),cons(X,XS)) -> mark(X) a__sel(s(N),cons(X,XS)) -> a__sel(mark(N),mark(XS)) a__take(X1,X2) -> take(X1,X2) a__take(0(),XS) -> nil() a__take(s(N),cons(X,XS)) -> cons(mark(X),take(N,XS)) mark(0()) -> 0() mark(2nd(X)) -> a__2nd(mark(X)) mark(cons(X1,X2)) -> cons(mark(X1),X2) mark(from(X)) -> a__from(mark(X)) mark(head(X)) -> a__head(mark(X)) mark(nil()) -> nil() mark(s(X)) -> s(mark(X)) mark(sel(X1,X2)) -> a__sel(mark(X1),mark(X2)) mark(take(X1,X2)) -> a__take(mark(X1),mark(X2)) - Signature: {a__2nd/1,a__from/1,a__head/1,a__sel/2,a__take/2,mark/1,a__2nd#/1,a__from#/1,a__head#/1,a__sel#/2,a__take#/2 ,mark#/1} / {0/0,2nd/1,cons/2,from/1,head/1,nil/0,s/1,sel/2,take/2,c_1/0,c_2/2,c_3/1,c_4/0,c_5/0,c_6/1,c_7/0 ,c_8/1,c_9/3,c_10/0,c_11/0,c_12/1,c_13/0,c_14/2,c_15/1,c_16/2,c_17/2,c_18/0,c_19/1,c_20/3,c_21/3} - Obligation: innermost runtime complexity wrt. defined symbols {a__2nd#,a__from#,a__head#,a__sel#,a__take# ,mark#} and constructors {0,2nd,cons,from,head,nil,s,sel,take} + Applied Processor: EmptyProcessor + Details: The problem is still open. MAYBE